Circular slide rule



March 6, 1951 H, HACHMUTH 2,544,224

CIRCULAR SLIDE RULE Filed Dec. 28, 1948 2 Sheets-Sheet 1 dlsJ FIG. INVENTOR.

K.H. HACHMUTH ATTORNEYS March 6, 1951 K. H. HACHMUTH 2,544,224

CIRCULAR SLIDE RULE Filed Dec. 28, 1948 2 Sheets-Sheet 2 m! N 00 2 a m 1K x. Ow

7' m 1 N 2' k L g ZL$ U I, Q "0 HVVENTUR. K.H,HACHMLVTH ATTORNEYS Patented Mar. 6, 1951 Karl H. Hachmuth, Bartlesville, Okla., assignmto lhillips Petroleum Company, a corporation of Delaware Application December 28, 1948, Serial No. 67,614

This invention relates of the slide rule type. In one of its morespecific aspects it relates to a calculating deviceof the sliderule type for use in solving equations relating to equilibrium between two phases in multicomponent systems. Ina still more specific as- 93 it relates to 81 slide rule for use insolving equations relating, to equilibrium vaporization or condensation of multi-component mixtures, especially of hydrocarbons.

alculator for use rinisolving equations relating to equilibrium betweenliqu'ld and vapor phases, two il-R d phases. liquid and solid phases, etc, in

ni'iuticomponeut syst m Another object or my v n 8 *0 provide ca ulator for solving etiuatlous relating to equilibrium vaporization of lgiul womronentmlxtures. especially of hydrocarbons.

Still another a l 0 y invention is nature wherein trial and error methods are in- 5 volved. a

Other objects and advanta es will be apparent :to those skilled in the art upon reading the following description which taken .in conjunction with the attached drawingrforms a part of this;

p ification.

In the era-w n: r18!!! 1 1S za'p'lan View of a preferred 'form of calculator oi vmy invention.

2 is a cross-sectional view :of the re-o rel-red term of calculator taken along the line o'tr'isure'i.

Figure 3 is a plan view of another embodiment of the calculator of my invention.

Figure a "is a cross-wsectlonal view of the embodiment of nay-calculator 0! Figure 3 and taken on the line 0-4 ofmiute. 3.

Referring to the drawing and specifically to Film: '1, H :re'ters. to a lower and larger 1'2 Claims. (Cl. 235-84) to a calculating device u Que object or my invention i to provide a circular disk on the periphery of which is marked a logarithmic scale l2. On topof this disk is disposed another circular disk l3 having a logarithmic scale I4 marked along its periphery. Disk I3 is of smaller diameter than disk II. The logarithmic scale I4 is so marked on the disk 13 that any point on this scale may be positioned adjacent any desired point on the peripheral scale 12 on the disk II. The disks H and I3 are pivoted at pivot point l5 in such a manner that one disk may be rotated with respect to the other. The pivot is preferably centrally located so that the circular disks are concentric.

On top of the disk I3 is an indicator or scale arm I6 which is also pivoted at point I5. A scale I I is marked along the radial edge I8 of the arm I6, and this edge l8 if extended would pass through the center or pivot point of the disks II and [13. The outer end of this indicator l6 may terminate as anoint or as a square end as illustrated in the drawing. While the scale on this indicator terminates some distance short of scale [2, the arm extends such a distance beyond this scale that the indicator may be used to indicate points on thisscale.

In a preferred embodiment the scales l2 and I4 are logarithmic scales from 0.1 or less to 100, the scales being arranged in reverse order with respect to each other Some curved lines l9 converging at a common point 2-0 extend to points on scale I4. These curves are functions of equilibrium vaporization constants K. The equilibrium constant K is the ratio of the mol fraction of a component in the vapor phase to its mol fraction in the liquid phase under equilibrium conditions at a given temperature and pressure. This constant K and its significance are well known in the art.

In a more general case, i. a, not limited to equilibrium :between a liquid and its vapor, these curves [9 are functions of an equilibrium constant defined as the ratio of the mol fraction of a component in a first phase to the mol fraction of that component in a second phase, assuming that a condition of equilibrium had been reached at given conditions of temperature and l9 (hereinafter'called the .K-linesi) is in which u degrees (using K=l.0 line as zero degrees); a=distance between the center or called the L-scale.

- 3 origin and the point of convergence of the K lines. This a value also determines the location of the L=1.0 point on the L-scale H of indicator l6; b is the length of scale [1; 7c=(Kl), in which K is the equilibrium constant; r is the distance from the center; C is the number of logarithmic cycles around the edge of the disk l3, which in this case is'3. The equation may be simplified by letting a=0, thereby putting the point 20 and the value L==l; at the center of the disks, and by letting b=,1. Ex-, pressing in radians, this K has the form 21 log curve equation then The K-curves may be laid out by making a series of concentric circles, with their centers '1,

at the center of the disk1l3 and with radii corresponding to the graduations on the indicator arm IS. The intersection of a radius (drawn from a graduation at the edge of disk l3 found from the function, K -LK +L=value of graduation) and a circle which represents the L value used in the function determines a point on the K curve whose K value is the same as used in the function. The terminals of the K value lines are divisions and subdivisions on a logarithmic (counterblockwise) scale M. For convenience, it is preferable to make the scale continuous, i. e., the-100 division identical with the 0.1 division on the 3-cycle circular scale I4.

The circular scale l2'on disk I l is also a 3-cycle logarithmic scale and is disposed clockwise around disk I l.

pen or" CIRCULAR SLIDE RULE I I2 on disk I l is termed the B-scale, the logarithmic scale M on the disk I3 is termed the U-scale 'while the scale 11 on the indicator or slide [6 is Multiplication Tomultiply 2x4, se t 2 on the U-scale opposite 'the numeral 4 on the B-scale and. read the answer 8 on the B-scale opposite 1.0 on the U-scale.

marked 10 may terminate as a pointer or For convenience, the graduation carry an arrow head for easy identification.

Division To divide 5 by 2, set the arrow on the U-scale opposite 5 on the B-scaleland read the answer ,2.5 on the B-scale opposite'the divisor 2 on the .U-scale.-

For numbers larger than 100 or smaller than 01, the'continuous scales B and U may be used.

Although not so marked on the drawing, each diivision corresponds to a series of numbers, 1000 fold apart from each other, for example, the division marked .2 simultaneously represents "0.000002, 0.002, 2, 2000, 2,000,000, and so on.

Hence any size number may be divided or multiplied on this device providing due regard is given to the proper positioning of the decimal point in the quotient or product.

Flash vaporization or'c'ondens'ation sected by the edge oitheindicato1; TB...

plied to a flash vaporization chamber operating at a known temperature and pressure, the compositions of the resulting vapor and/or liquid streams can be calculated from the equation:

"in whichxac, Xbc, Xcc, etc., are the concentrae s ein which XL stilema t er cent of, s component leaving the vaporizatioii'lchamber h nthei'li'qu'id phase. Since the composition of thefeed stream and the conditions of temperatui andpre'ssur'e are known, the values Xe and known for each component; .By substituting v Equationj these known values X5 and K',' a d ain assumedor estimated value of V, onefcarf fs'olvethe equation for XL. When this calculation has been made' -for all the components of the feed stre ar r i, the values of XI. are added lithe sum is'gre'aterp'r less than a different 'va tie of'V must-'be'assumed. These calculations are repeated 'until' the proper value of V is found, i. e. un til Equation l'issatisfied. It is obvious that sucl'i'ca'lculat n'sare tinie consuming and tediousif 'they a-re oa'r i'edout in the manner juStdescribed-ftli'at'is by substituting in Equation 2 and solving al'g ebraically for-XL. The present invention, ho'w'ever, providesasimple and rapid method- 0f solving for X5, -there'by reducing to a minimum the *time r'equired forsolu tion of Equation 1 ;--Using--the circular slide 'r ule of this invention as 'it is illustratedm. Figure .1, i. e., when the. scale I1 is 'graduatedainterms'oi mol fraction of the feed which remains asiliquid (L) rather than in terms ;of.vapor.-...(V); itis first necessary to convert the assumedsvalueeof;V to an equivalent value in terms of L .;;f This is readily accomplished from the relationship V+L=l00 (mol percent) 7 Q1' V +Ll (mol fraction}.

The logarithmic scale 12 (B-scale) ongtlie larger disks ll represents walues pf Xe or values of XL, the concentration in mol fraction pr -in mol per cent of a given; component the charg (X or. nih qu drha e KP T F The calculation of XL is then c rri'ed 0utIby set ting the arrowed line 10 of the u-sai'bpiidsite the known value or Xe onthetB' s ingthe scale arm l6 until thel'value "of fL eduivaleht to the assumed? value of" Vfc'oincides' withrthe known value of K and rea'ding" the answer X1. from the B-scale'at the'point where Q ample, X0 is 18.2, andll is. assumedtb be '.30, ,vv.ith K as 5, X1. is found -tobe 4 7$. Asimilarcalculation is carried out for. each component,substituting the known 'valuesoi Xe, and K, and'using th 1 same assumed value bin-=30, until the Xr. values io ia ,thepm p m. ing i fisxplinid above, if the sum of the calculate'dxr'fvaliies 1s 100, then the assumed value of. i. i the eerrect value and the composition of the liquid ieavms the va orization chamber is given by the values of XL. If the sum of the several XL values is less than or is greater than 100, thena different value for L is assumed and the calculations repeated. This procedure is repeated until an L value is found which yields XL values, thesum of which equals 100, or in other words satisfies Equation 1. 7

Test for dew point of a mixture A mixture of the following composition afloat conditions of temperature and pressure aresuen that the equilibrium vaporization constants (K values) are as given below; to determine whether 3 the-mixture is above, below 01 at its deW poi-mi COMPOSITION Oompohent Pei-,ceiit K go Divide on the slide rule each Xe by its corresponding K as described above for simple division. These quotients are as follows:

Component Xe/K I Sun1 L .4, H25

Test for boiling point of a, mixture Using the same Xe and K values as given in the dew point calculation, multiply each Xc value by the corresponding K value to produce X'cK products, as follows:

Component X.=K

If the sum of the caleulated-XK'values is greater than'100 the mixture is above its boiling point '70 and partly or entirely vaporized; if the sum or these calculated values e eal sroo, the mixture is at its boiling point while "if the sum of these calculated values is less than 100. the mixture is below its boiling point; In this example-"the "7' 21K vaiue for component A was greateHn-sn 1'00 therefore it was unnecessary to carryout-the multiplications for the other components. In this-example since the mixture is above its boiling point it is evident that the mixture is prescut as all vapor or part vapor. and part liquid.

To find fraction of Mature existing as liquid Using the same mixture composition and K values as used above in the dew point and boiling point calculations, the mol fraction of the mixtime-existing as liquid is found as follows:

V Galculated XL Values for Assumed Cd M01 IL Values I m- W .4. a. l a... otent g g- K .1 V

Thai No. 1 Trial 'No. '2 (for L s) (ior 1.;

A-- 10 s. 5 1. 4 4. 0*) B- 5 7.9 6.5 6.1 Q 1.1 4.7 4.6 4.55 D 5 29.5 34. 7 37. 4 E4 .1 113.2 31 2- 4s. s

The above tabulated trial 1, trial 2', etc., calculations were obtained asfollows: 7

Assume some mol fraction of the mixture to existas liquid, for example 0.3-. Set pointer (1.0 on U-scale) opposite 10 on B scale, movev indicator until the 0.3 division on the L-scale exact ly intersects the K=curve labeled 10, and read 5.5 on the B-scale and record this value in the trial-No. 1 column for component A. For component B, set pointer on U-scale opposite 30 on B-scale, rotate indicator I6 until .30 on L-scale coincides with the K-curve marked 5, read 7.9 on the B-scale and enter in the trial-1 column for component B. For component C, set pointer of U-soale opposite 5 on B-scale, rotate slide until .30 on L-scale' coincides with the K-line marked 1.1., read 4.7 on the B-scale and enter in its proper space in the trial-1 column. Upon carrying out the corresponding calculations for components 13 and E the values 295 and 16.2, respectively, are found and are recorded in their proper spacesv in the trial No. 1 column. Upon adding the calculated XL values in trial No. 1 column, the sum is 63.8, and since this sum is not 100, another value of L must be assumed and similar calculations made.

For the second trial, assume a value of 0.1 for L, and make, calculations similar to those just described, but using the 0.1 division on the L- soale to coincide with the respective K-curves. Results of this trial are recorded in the trial N9. 2" column. Thevalues in this column total 82.4.

For a third trial, assume a value of 0.02 for L, and make calculations similar to those just made, excepting use the value 0.02 on the L- scal'e. These calculated values total 10062.

For many purposes this total of 100.62 may be sufilciently near 100. However, the sum of XL values was not exactly and therefore the mol fraction of liquid was not exactly 0.02.

Because the K-curves approach parallelism with the periphery of the B-scale at low K values, the position at which a given value on the L-scale matches a chosen K valuebecomes difficult to determine. Assuming the intersection of the L- 'scale. with the chosen K-curve may be in error by as much -as--$.0.01--the possible-variatiom-in reading the X1. value for components D and E is given in the third column of the following table:

Possible variation in XL Possible var- These variations represent only :one per cent in the fraction remaining liquid but such large errors in the composition of the residual liquid are annoying. A much higher degree of precision may be obtained by using a different set of curves drawn on disk l3 in place of the low value K-curves and designated as S-curves" 2|. The possible variation when using the S-curves is given in the fourth column in the above table. For component D, when using the S-curve, the error in setting is about the same as when using the K=0.5 curve. For component E, when using the S-curve the error in setting is only about one tenth that found when using the K=0.1 curve. When K-values are less than 0.5, the S-curves may be used to increase accuracy of calculations involving K values.

The S-values are the sum of the K values and the fraction L remaining as liquid, as represented by the equation S=K+L. By the nature of the S-curves the numerical value of L and the corresponding value of K both lie along the straight line of the L scale. For example, in S=K+L, if S is .7, as far as S=.7 is concerned it is immaterial whether L is .l or .6, or whether K is .6 or .1; when L=.6 is set on the S=.'l curve, this setting is slightly difficult since the S=-.7 curve is not far from parallel to the L-scale at this point. The 8:.7 curve intersects the L-scale at another point, viz. at the point 0.1, and at this point of intersection the 8:07 curve is less nearly parallel to the L-scale than it is at the L=.6 point, hence a more accurate reading on the B-scale may be made in this case if the value 0.1 on the L-scale-is used to determine the posi tioning of the L-scale than if the point L=.6 on the L-scale is used. Thus the'smaller value of K or L should be used on the L-scale to be able to determine more accurately its setting with respect to the S-curve. The smaller numbers on the L-scale are nearer the outer end of the scale and hence make possible more accurate settings of the L-scale on the S-curve.

In the problem illustrated above, the choice of L=0.02 gave a sum of 100.62 for the XLS- Rather than make another trial, the true value of L may be determined by interpolation from the L=0.1 and the L=0.02 X1. values:

10082.4 (Estnnated)L-0.l 0.1 0.0a Upon solution of this equation L is found to be 0.023, and the true XLS found by interpolation are as follows:

Component The difierence in composition between the estimated (trial No. 3 for L=-.02) and the true values is insignificant. The difierence in the Ls (L=.02 by my slide rule and L=0.023 by interpolation) might be significant since it amounts to about 15% of the liquid residue. However, the precision of the slide rule is not adequate to determine the L any closer than that obtained. Further, the K (equilibrium vaporization) values used are never sufiiciently accurate to justify more precise calculations. For instance, most generalized K-charts cannot be read with better precision than about i2% of the K. A change in K from 0.098 to .102 for component E will vary the corresponding XL from 49.0 to 47.5, which variation is twice as great an uncertainty as that with which the rule can calculate the answers. In other words the precision of my slide rule is better than the K charts and therefore use of the slide rule will not introduce uncertainties in the results greater than those already present from other sources.

Using the interpolated values to complete the problem:

Amount of Residual Compo- Com ggg a i g Feed Vapor, position of qui I (One Mols poi Residual ponent Res dual Mols per M Ole) M 01 of vapor Llqmd' XL g j Feed Molpcrcent A 4. 04 000929 .40 399071 40. B 0. 08 .001514 30 298480 30. 50 O 4. 52 001040 05 048900 5. 01 D 37. 13 008640 19 .181360 18.57 E 48. 23 011093 .00 048907 5. 0l

The values of the Amount of residual liquid column were obtained by dividing the corresponding composition value in the Composition of residual liquid, X1. column by and multiplying by 0.023. The sum of these products should then equal 0.023. These calculated products are subtracted from the respective amount of each component in one mole of feed to give the amount of each component in the Residual vapor moles per mole of feed column. The last column in the above tabulation gives the composition of the residual vapor as calculated from the quantities of the individual components left in the vapor.

A check for mistakes and precision may be made by dividing the individual values in the residual vapor composition column by the corresponding K5 and comparing with the X16:

Y Residual Vapor Component Compo- K Y/K X1.

sition, Mole percent The disagreement between the above Y/K value'for component E and the corresponding X1. value is mainly due to difliculty in reading the slide rule for small K values and in a possible slight variation in the value 0.023 which probably is not exactly 0.023000. However, the degree of precision is adequate for all ordinary cases.

iSjIhe polar; coordinate equation oi thc swcurves Spaces between lines for constant K-values or S-values may preferably be subdivided so that no space between subdivisions shall .beggreater than degrees of arc (with center at center of circle) nor smaller thanabout 1 degree of arc. The S-curves shoul'd 'preferably be continued as far as the K=0I01 or zero curveis continued, i. e., to the number "ID point on the (J-scale.

While the above description is directed to the construction and use cf a circular "slide rule my invention is not to be-limited to this type ofslide rule although it is the preferred type. A straight or linear type of rule may be *made' embodying m-y-invention. I

Figures 3 and 4 illustrate the construction of this type of rule. Reference numeral iirefers to the lower body member and reference numeral 3 l a to an upper-body member. body members are provided wither-coves to accommodate a movable slide 32. Another groove is provided in the lower body "member to accommodate the small slide anchor '38. n maybe fastened to theanchor 'li bysome screws "8!.

To this slide anchor isgiastenedpas by screws. an indicator or arm in which is provided y with a linerscale 43. 'lheupperendlowerbody meinbers may be rigidly "fastened together by braces. 34- with the aid 0! screws 85. .Thelower bodyv member 3! is provided within. 3-cycle-losarithmic. scale 31 while slide '32 is also provided with a 3-cycle logarithmic scale '36. Scales ii, and 31 are exactly the same length.areexactly-identical. but reversed.

Fromscale point 1.0 on the slide scale ills drawn a straight line 44, as shown, which terminates at point. K' curves II are logarithmic cartesian coordinate curves plotted from the}! and Y e'quaions.

X= a 1ogio[Y+K(1-Y)'l a equals the length of the logarithmic cycle. 1 These K" curves 'mayelso be plotted fromvalue taken from the corresponding curves'o'f Figure 1. The S curves 4; are also logarithmic carteslan coordinate curves plotted on slide 32 with the point 40 as the origin and terminating at points along the scale 36 from 1.0 to 0.1 and may be constructed rfomthesequations This X equationisliound bysubotituting the ,equivalent of K, which "is is -L), ior K infthe above equation:X= L-l-K( 1 -"L). Y The 5 curves 42, may-also begplottedfon the slide-:lI-irom-points taken from the securves of I41igures.1

; The operation of the rule-of Figure -3 #is substantially the some as for the circular-rule :of Figure 1.- Simple multiplication and; division operations are madeusing the scales 36 and 3'1,

as'with conventional slide rules.

incides with a K-curve or an S-curve for the component in question and read the answer on the logarithmic scale 31.

In the linear slide rule embodiment of my invention the S'-curves ofier only slight advantage over use of K'-curves, and accordingly the S'- curves may be omitted.

My slide rule, that is, either embodiment may be constructed of materials ordinarily used for constructing such instruments, that is, wood,-'

aluminum, magnesium, Celluloid, or the like.

It will be obvious to those skilled in the art" that many variations and alternations in the rules herein disclosed may be made, such as using 4 cycle log scales in place of 3 cycle scales when large K values are involved, or even'using 2 cycle log scales when K values are not above 10, andyet remain within the intended spirit and scope of my invention.

Having disclosed my invention, I claim: 7

1. A slide rule for use in solving equations'relating to equilibrium between two phases of a system consisting of a plurality of components comprising in combination a first member having a logarithmic scale and a series of curves thereon, said curves converging at a common point [on said member and terminating at points-on said scale, each curve of said series of curves beinga function of the equilibrium K-constant, K =y/x, in which 3/ is the mol fraction of one componentin a 'first phase and m is the mol fraction oi ai 1 component in a. second phase, a second member slidable relative to the first member and having a second logarithmic scale inverted with respect to the first logarithmic scale, and a slide' member having a linear scale disposed normal to said'first and second scales and intersecting said curves and said secondscale 2. A slide rule for use" in solving equations relating to equilibrium between two phasesof a system consisting of a plurality 'of components comprising in combination a first member having a logarithmic scale and a first series of curves thereon, said curves converging at a common point on said first member and terminating at points on said scale, each curve of said series of curves being a function of the equilibrium K constant, K=y/w, in which y is the mol fraction of one component in a first phase and a is the mol fraction of said component in a second phase,

, :a: second series of curves thereon, each curve of .said second series of curves also terminating at points on said scale and being a second function oi? said equilibrium K-constant of said component, a second member slidable relative to the first member and having a second logarithmic scale inverted with respect'to the first logarith mic scale, and a slide member-having a linear scale disposed normal to said first and second scales and intersecting said curves and said sec-- ond scale. I r

3. A circular slide rule for use in solvmg equa i ationsrelating to equilibrium between two phases of curves being a function of the equilibrium K constant, K=y/:c, in which y is the mol fraction of one component in a first phase and'm is the mol fraction of said component in a second phase, a second circular member rotatable relative to the first circular member and having a second logarithmic scale inverted with respect to the first logarithmic scale, and a slide member having a linear scale disposed normal to said first and second scales and intersecting said curves and said second scale, said points of said intersection defining a straight line passing through the center of rotation of said slide member.

4. A circular slide rule for use in solving equations relating to equilibrium between two phases of a system consisting of a plurality of components comprising in combination a first circular member having a logarithmic scale disposed around the periphery thereof and a series of curves thereon, said curves converging at a common point on said member and terminating at points on said scale, each curve of said series of curves being a function of the equilibrium K- constant, K:y/:r, in which 11 is the mol fraction of one component in a first phase and 1c is the mol fraction of said component in a second phase, a second circular member rotatable relative to the first circular member and having a second logarithmic scale disposed adjacent said lfirst logarithmic scale, said first logarithmic scale disposed inverted with respect to said second logarithmic scale, and a slide member having a linear scale disposed normal to said first and sec ond scales and intersecting said curves and said second scale, said points of said intersection defining a straight line passing through the center of rotation of said slide member.

5. The circular slide rule of claim 4 wherein the logarithmic scales are 3-cycle scales.

6. The circular slide rule of claim 4 wherein each curve of said series of curves is defined by the equation in which =degrees using the K=1 line as zero degrees, a=distance between the center and a 1.0 graduation on said linear scale, b=the length of said linear scale, k: (K 1), r=distance from the center and C=the number of logarithmic cycles of said logarithmic scales.

7. A circular slide rule for use in solving equations relating to vaporization equilibrium between two phases of a system consisting of a plurality of components comprising in combination a first circular member having a logarithmic scale disposed around the periphery thereof, and a first series of curves thereon, said curves converging at a common point on said member and terminating at points on said scale, each curve of said series of curves being a function of the equilibrium vaporization K-constant, K :y/x, in which 11 is the mol fraction of one component in the vapor state and a: is the mol fraction of said component in the liquid state, and a second series of curves thereon, said curves of said second series of curves also terminating at points on said scale and being second functions of said K-constant of said component, asecond circular member rotatable relative to the first circular member and having a second logarithmic scale disposed adjacent said first logarithmic scale and inverted with respect thereto, and a slide member having a linear scale disposed normal to said first and second scales and intersecting said curves and said second scale, said points of said intersection defining a straight line passingthrough the center or rotation of said slide member.

8. The circular slide rule of claim '7 wherein each curve of said first series of curves is defined by the equation e e-glogpimq in which 0'=deg'rees using the K=l line as zero degrees, a=distance between the center and a 1.0 graduationon said linear scale, k=(K-l), r=distance fromthe center and C=the number of logarithmic jcycles comprising first said logarithmic scale, and each curve of said second series of curves is defined by the equation in which 0=degrees using the K =1 line as zero degrees, zr=distance between the center and a 1.0 graduation on said linear scale, r=distance from the center, C'=the number of logarithmic cycles comprising first said logarithmic scale, and S :K +L in which K is the equilibrium vaporization constant of said one of said components and L is a numerical value from said linear scale.

9. A linear'slid'e rule for use in solving equations relating to equilibrium vaporization oi a system consisting of a plurality of components comprising in combination a first linear member having a logarithmic scale and a series of curves thereon, said curves converging at a common point on said first linearmember and terminating atjpo'ints on said scale, each curve of said series of curves being a function of the equilibrium vaporization K -con'stant,' K :y/zv, in which 11 is the mol fraction of one component in the vapor state and a: is the mol fraction of said component in the liquid state, a second linear member slidable relative to the first and having a second logarithmic scale inverted with respect to the first logarithmic scale, and a slide member having a linear scale disposed normal to said logarithmic scales and intersecting said curves and said second scale.

10. The linear slide rule 01. claim 9 in which the two logarithmic scales are 3-cycle scales and the curves of said first series of curves are defined by the equations and the curves of said second series 01' curves are defined by the equations X=a-10g1o [S'--YS'+Y 11. A linear slide rule for use in solving equations relating to equilibrium between two phases of a system consisting of a plurality of components comprising in combination a first linear member having a logarithmic scale and a series of curves thereon, said curves converging at a common point on said first linear member and terminating at points on said scale, each curve ofsaid series of curves being a function 01 the equilibrium K-c'onstant, K=y/x,' in which y is the mol fraction of one'component in a first phase and a: is the mol fraction 01' said component in a second phase, and a second series of curves thereon, said curves of said second series of curves also terminating at a point on said scale.

each curve 01' said second series of curves being a function of the equilibrium S-constant, S=K+L, iI1 'WhiCh K is the above mentioned K- constant and L is a numerical value from a slide member having a linear scale normal to said logarithmic scale as subsequently mentioned, a second linear member slidable relative to the first and having a second logarithmic scale thereon and inverted with respect to the first logarithmic scale, and a slide member having a linear scale disposed normal to said logarithmic scales and 10 KARL H. HACHMUTH.

REFERENCES CITED The following references are of record in the file of this patent:

UNITED STATES PATENTS Number Name Date 1,157,609 Backstrand et al Oct. 19, 1915 2,443,882 Allen June 22, 1948 FOREIGN PATENTS Number Country Date 292,223 Great Britain June 14, 1928 359,573 Italy Mar. 10, 1938 OTHER REFERENCES Petroleum Refinery Engineering, by W. L. Nelson, p. 230, published byMcGraw-Hill Book Co., Inc., N. Y., 1941. 

